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\section{Problem Statement}\label{sec:prob}

\begin{table*}[htpb]
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\centering
\caption{Notations in this paper}\label{tbl:notation}
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\begin{tabular}{|>{\columncolor[gray]{.85}}L{3cm}|L{4cm}|>{\columncolor[gray]{.85}}L{4cm}|L{4.5cm}|}
\hline
$\mathcal{T}$ & Target set &$\mathcal{S}$ & Auxiliary set\\
\hline
$\mathcal{X}_{T}$ & Target item set & $\mathcal{X}_{S}$ & Auxiliary item set\\
\hline
$\mathcal{Y}_{T}$ & Target cluster space & $\mathcal{Y}_{S}$ & Auxiliary cluster space\\
\hline
$\mathcal{F}$ & Common feature set & $\mathcal{Y}_{F}$ & Feature cluster space\\
\hline
$h_{T}: \mathcal{X}_{T}\rightarrow\mathcal{Y}_{T}$ & Target clustering function & $h_{S}: \mathcal{X}_{S}\rightarrow\mathcal{Y}_{S}$ & Auxiliary clustering function\\
\hline
$h_{F}: \mathcal{F}\rightarrow\mathcal{Y}_{F}$ & Feature clustering function
& $C_{\mathcal{F}}$ & Number of clusters of $\mathcal{F}$\\
\hline
$N_T$ & Number of instances in target set & $N_S$ & Number of instances in auxiliary set\\
\hline
$C_T$ & Number of clusters in target set & $C_S$ & Number of clusters in auxiliary set\\
\hline
$\mathcal{J}$ & Objective function & $D(p\|q)$  & KL divergence\\
\hline
$\tilde p(\mathcal{X},\mathcal{F})$ & Joint probability distribution of $\mathcal{X}$ and $\mathcal{F}$ on co-clusters $\mathcal{Y}_{X}$ and $\mathcal{Y}_{F}$ & $\lambda_f$ & Influence of auxiliary set for feature $f$\\
\hline
$\mathfrak{D}_f$ & Similarity of feature between $\mathcal{X}_{T}$ and $\mathcal{X}_{S}$ & $w$ & weight of feature in a data set\\
\hline
%Three & Four\\
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\end{tabular}
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In this section, we describe the problem we focus on in this study and introduce some notations and definitions used in this paper.

A target set $\mathcal{T}=\{\mathcal{X}_{T},\mathcal{Y}_{T}\}$ has $\mathsf{x}_{t1},\ldots,\mathsf{x}_{tN_{T}}\in\mathcal{X}_T$, where $\mathcal{X}_{T}$ is the target item set of $N_{T}$ testing instances and the corresponding cluster space $\mathcal{Y}_{T}=\{\mathsf{y}_{t1},\mathsf{y}_{t2},\dots,\mathsf{y}_{tC_{T}}\}$ with possible $C_{T}$ partitions. An auxiliary set $\mathcal{S}=\{\mathcal{X}_{S},\mathcal{Y}_{S}\}$ consists of $\mathsf{x}_{s1},\ldots,\mathsf{x}_{sN_{S}}\in\mathcal{X}_S$, where $\mathcal{X}_S$ is the auxiliary item set of $N_{S}$ training examples, with the corresponding cluster space $\mathcal{Y}_{S}=\{\mathsf{y}_{s1},\mathsf{y}_{s2},\dots,\mathsf{y}_{sC_{S}}\}$ having possible $C_{S}$ partitions. In this study, we focus on transfer clustering, which is defined as follows.

\begin{mydef}
(Transfer Clustering)
Given a target set $\mathcal{T}$ and an auxiliary set $\mathcal{S}$, transfer clustering investigates searching for a predict clustering function $h_{T}: \mathcal{X}_{T}\rightarrow\mathcal{Y}_{T}$ to cluster $\mathsf{x}_{t}$ that are similar to the same partition while those are more dissimilar to different clusters with the assistance of $\mathcal{X}_{S}$, where there is no information about $\mathsf{y}_{t}$ and $\mathsf{y}_{s}$ of any $\mathsf{x}_{s}\in\mathcal{X}_{S}$ and $\mathsf{x}_{t}\in\mathcal{X}_{T}$. 
\end{mydef}
We try to cluster the target and the auxiliary data simultaneously to allow the auxiliary data to influence the target data through a common feature set $\mathcal{F}$ with possible $C_\mathcal{F}$ partitions. Let $h_{S}: \mathcal{X}_{S}\rightarrow\mathcal{Y}_{S}$ and $h_{F}: \mathcal{F}\rightarrow\mathcal{Y}_{F}$ represent the clustering functions in auxiliary data $\mathcal{S}$ and feature set $\mathcal{F}$ respectively.  \comment{$\widetilde{\mathcal{X}_T}$, $\widetilde{\mathcal{X}_S}$ and $\widetilde{\mathcal{F}}$ represent the clustered $\mathcal{X}_T$, $\mathcal{X}_S$ and $\mathcal{F}$ separately.}
Some notations and terminologies are listed in Table~\ref{tbl:notation}.

